The ex-Arsenal midfielder, who held talks with Newcastle last season, is available after leaving Lyon in 2014 to take a break for personal reasons.
He spent three years in charge at the Stade de Gerland and won the Coupe de France in 2012 having previously won the Premier League in 1998 as a player with the Gunners.
Garde is odds-on favourite with bookmakers to become the next Villa manager, while former Leicester boss Nigel Pearson, Brendan Rodgers, available after leaving Liverpool, and David Moyes, currently in charge at Real Sociedad, have also been backed.
Sherwood was sacked on Sunday after Villa plummeted to the bottom of the Barclays Premier League on the back of six straight defeats
Press Association Sport understands the 46-year-old will net a 12-month pay-off after leaving just eight months into a three-and-a-half-year deal
He was axed along with assistant Ray Wilkins, first-team coach Mark Robson and performance analyst Seamus Brady following Saturday's 2-1 defeat to Swansea.
The club said in a statement: "The board has monitored the performances closely all season and believes the results on the pitch were simply not good enough and that a change is imperative.
"However, the club would like to place on record its sincere thanks to Tim for all his efforts during a difficult period last season and for the many positive contributions he has made to the entire football set-up during his time with the club
We wish him well in the future."
Villa's Under-21 boss Kevin MacDonald has been placed in caretaker charge and is expected to manage the side for Wednesday's Capital One Cup trip to Southampton as well as next Monday's league game at Tottenham.
MacDonald was previously caretaker boss in 2010 after Martin O'Neill quit and was brought back to the club by Sherwood in February after a coaching spell at Nottingham Forest.
Source : PA